Definition

$(a_{n}) = f : N \to R$ $a_{m} = f (m)$

A function from natural numbers to the element at each position. The notation of a sequence can be generalized to an indexed family.

Subsequence

$(a_{n})_{n \in N} = (a_{n_{k}})_{k \in N}$ , where $(n_{k})_{k \in N}$

A sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements

$(n_{k}) \forall k \in N, n_{k} \leq n_{k + 1}$

$(n_{k}) \forall k \in N, n_{k} \geq n_{k + 1}$

$(n_{k}) \forall k \in N, n_{k} < n_{k + 1}$

$(n_{k}) \forall k \in N, n_{k} > n_{k + 1}$

If $\exists M > 0, \forall n \in N, ∣ a_{n} ∣ \leq M$ , then the sequence $(a_{n})$ is said to be bounded.